Question:
Find the average of even numbers from 4 to 1324
Correct Answer
664
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 1324
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 1324 are
4, 6, 8, . . . . 1324
After observing the above list of the even numbers from 4 to 1324 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 1324 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 1324
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 1324
The average of the numbers forming an Arithmetic Series
= The first term (a) + The last term (ℓ)/2
⇒ The average of numbers forming an Arithmetic Series = a + ℓ/2
Thus, the average of the even numbers from 4 to 1324
= 4 + 1324/2
= 1328/2 = 664
Thus, the average of the even numbers from 4 to 1324 = 664 Answer
Method (2) to find the average of the even numbers from 4 to 1324
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 1324 are
4, 6, 8, . . . . 1324
The even numbers from 4 to 1324 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 1324
The Average of the given numbers
= Sum of the given numbers/Total number of given numbers
Thus, to find the average of the given numbers, first, we need to find their sum and the total number of given numbers
Finding the number of terms
For an Arithmetic Series, the nth term
an = a + (n – 1) d
Where
a = First term
d = Common difference
n = number of terms
an = nth term
Thus, for the given series of the even numbers from 4 to 1324
1324 = 4 + (n – 1) × 2
⇒ 1324 = 4 + 2 n – 2
⇒ 1324 = 4 – 2 + 2 n
⇒ 1324 = 2 + 2 n
After transposing 2 to LHS
⇒ 1324 – 2 = 2 n
⇒ 1322 = 2 n
After rearranging the above expression
⇒ 2 n = 1322
After transposing 2 to RHS
⇒ n = 1322/2
⇒ n = 661
Thus, the number of terms of even numbers from 4 to 1324 = 661
This means 1324 is the 661th term.
Finding the sum of the given even numbers from 4 to 1324
The sum of all terms (S) in an Arithmetic Series
= n/2 (a + ℓ)
Where, n = number of terms
a = First term
And, ℓ = Last term
Thus, the sum of all terms (S) of the given even numbers from 4 to 1324
= 661/2 (4 + 1324)
= 661/2 × 1328
= 661 × 1328/2
= 877808/2 = 438904
Thus, the sum of all terms of the given even numbers from 4 to 1324 = 438904
And, the total number of terms = 661
Since, the average of the given numbers
= Sum of the given numbers/Total number of given numbers
Thus, the average of the given even numbers from 4 to 1324
= 438904/661 = 664
Thus, the average of the given even numbers from 4 to 1324 = 664 Answer
Similar Questions
(1) Find the average of the first 206 odd numbers.
(2) Find the average of the first 4626 even numbers.
(3) Find the average of the first 3728 odd numbers.
(4) Find the average of odd numbers from 11 to 1191
(5) Find the average of odd numbers from 13 to 505
(6) Find the average of the first 2233 even numbers.
(7) Find the average of the first 2002 odd numbers.
(8) Find the average of odd numbers from 3 to 617
(9) What is the average of the first 1554 even numbers?
(10) Find the average of even numbers from 4 to 240